Find the Derivative - d/dx y=x square root of 4-x

Problem

d()/d(x)*x√(,4−x)

Solution

1. Identify the rule needed for the expression, which is a product of x and √(,4−x) We must use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Rewrite the square root as a power to make differentiation easier.

y=x*(4−x)(1/2)

3. Apply the product rule where u=x and v=(4−x)(1/2)

d(y)/d(x)=xd(4−x)/d(x)+(4−x)(1/2)d(x)/d(x)

4. Differentiate the first term using the chain rule.

d(4−x)/d(x)=1/2*(4−x)(−1/2)⋅(−1)

5. Differentiate the second term.

d(x)/d(x)=1

6. Substitute these derivatives back into the product rule expression.

d(y)/d(x)=x⋅(−1/(2√(,4−x)))+√(,4−x)

7. Simplify the expression by finding a common denominator, which is 2√(,4−x)

d(y)/d(x)=(−x)/(2√(,4−x))+(2*(4−x))/(2√(,4−x))

8. Combine the numerators.

d(y)/d(x)=(−x+8−2*x)/(2√(,4−x))

9. Combine like terms in the numerator.

d(y)/d(x)=(8−3*x)/(2√(,4−x))

Final Answer

(d(x)√(,4−x))/d(x)=(8−3*x)/(2√(,4−x))

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